Determine which of the following polynomials has a factor :
i)\(x^3 + x^2 + x + 1\)
ii)\(x^4 + x^3 + x^2 + x + 1\)
iii)\(x^4 + 3x^3 + 3x^2 + x + 1\)
iv)\(x^3 - x^2 - (2 + \sqrt{2})x + \sqrt{2}\)

The zero of x + 1 is –1.


i)Let \(p(x) = x^3 + x^2 + x + 1\)
\(\therefore \) To check, whether x + 1 is a factor of \(x^3 + x^2 + x + 1\).

By factor theorem,
\(\therefore \) \(p(–1) = {–1}^3 + {–1}^2 + {-1} + 1\)
\(\Rightarrow \) \(p(–1) = –1 + 1 -1 + 1\)
\(\Rightarrow \) \(p(-1) = 0\)

Hence, x + 1 is the factor of \(x^3 + x^2 + x + 1\)


ii)Let \(p(x) = x^4 + x^3 + x^2 + x + 1\)
\(\therefore \) To check, whether x + 1 is a factor of \(x^4 + x^3 + x^2 + x + 1\).

By factor theorem,
\(\therefore p(–1) = (-1)^4 + (–1)^3 + (–1)^2 + (-1) + 1\)
\(\Rightarrow \) \(p(–1) = 1 – 1 + 1 - 1 + 1\)
\(\Rightarrow \) \(p(-1) = 1\)

Hence, x + 1 is not the factor of \(x^4 + x^3 + x^2 + x + 1\)


iii) Let \(p(x) = x^4 + 3x^3 + 3x^2 + x + 1\)
\(\therefore \) To check, whether x + 1 is a factor of \(x^4 + 3x^3 + 3x^2 + x + 1\).

By factor theorem,
\(\therefore p(–1) = {-1}^4 + 3{–1}^3 + 3{–1}^2 + {-1} + 1\)
\(\Rightarrow p(–1) = 1 – 3 + 3 - 1 + 1\)
\(\Rightarrow p(-1) = 1\)

Hence, x + 1 is not the factor of \(x^4 + 3x^3 + 3x^2 + x + 1\)


iv) Let \(p(x) = x^3 - x^2 - (2 + \sqrt{2})x + \sqrt{2}\)
\(\therefore \) To check, whether x + 1 is a factor of \(x^3 - x^2 - (2 + \sqrt{2})x + \sqrt{2}\).

By factor theorem,
\(\therefore p(–1) = {–1}^3 - {–1}^2 - (2 + \sqrt{2}){-1} + \sqrt{2}\)
\(\Rightarrow p(–1) = – 1 - 1 + 2 + \sqrt{2} + \sqrt{2}\)
\(\Rightarrow p(-1) = 2\sqrt{2}\)

Hence, x + 1 is not the factor of \(x^3 - x^2 - (2 + \sqrt{2})x + \sqrt{2}\)